Derivative of \( \displaystyle \frac{\ln{\left(\sin{\left(3 x - 1 \right)} \right)}}{3} \)
Problem 2.1895 · hard
Differentiate \( \displaystyle f(x) = \frac{\ln{\left(\sin{\left(3 x - 1 \right)} \right)}}{3} \).
- \[ \frac{d}{d x} \frac{\ln{\left(\sin{\left(3 x - 1 \right)} \right)}}{3} \]derivativeStart with the derivative of the function.✓ Proved
- \[ = \frac{\frac{d}{d x} \ln{\left(\sin{\left(3 x - 1 \right)} \right)}}{3} \]constant-multiplePull out the constant factor 1/3.✓ Proved
- \[ = \frac{\frac{d}{d x} \sin{\left(3 x - 1 \right)}}{3 \sin{\left(3 x - 1 \right)}} \]chainApply the chain rule for the logarithm.✓ Proved
- \[ = \frac{\cos{\left(3 x - 1 \right)} \frac{d}{d x} \left(3 x - 1\right)}{3 \sin{\left(3 x - 1 \right)}} \]chainApply the chain rule to the sine function.✓ Proved
- \[ = \frac{\cos{\left(3 x - 1 \right)}}{\sin{\left(3 x - 1 \right)}} \]derivative simplifyDifferentiate the inner linear function. Simplify the expression by canceling the 3 and 1/3.✓ Proved
- \[ = \cot{\left(3 x - 1 \right)} \]simplifyUse the cotangent identity.✓ Proved
Answer \( \frac{1}{\tan{\left(3 x - 1 \right)}} \)
Mind the domain. The answer is also defined at points where f(x) is not. Substituting there gives a number that is not a slope of f.
Every line of this solution was proved by the computer algebra system SymPy. The answer was also checked a second way, without looking at the solution. The reviewers disagree about how one step is explained; every verdict is in the receipt.
The full receipt
| Line | Status | Checked by | Detail |
|---|---|---|---|
| 1 | ✓ Proved | sympy 1.14.0 | line 1 is the problem as stated |
| 2 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments |
| 3 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where sin(3*x - 1) = 0 |
| 4 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where sin(3*x - 1) = 0 |
| 5 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where sin(3*x - 1) = 0 |
| 6 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where sin(3*x - 1) = 0 |
| 7 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where sin(3*x - 1) = 0 cot has poles at multiples of pi |
| answer | ✓ Proved | sympy 1.14.0 | final line against the stated answer: simplify(a - b) reduced to 0 tan has poles at odd multiples of pi/2 undefined where tan(3*x - 1) = 0 |
| answer, a second way | ✓ Proved | sympy 1.14.0 | SymPy differentiated f directly and got the stated answer |
Reviewers
gpt-oss:20b: passqwen3.6:27b-mlx: pass
Every verdict on record (4)
gpt-oss:20b: pass 2026-10-09qwen3.6:27b-mlx: pass 2026-10-09qwen3.6:27b-mlx: pass 2026-10-09gpt-oss:20b: fail (error) 2026-10-09 — Step 4 applies two rules at once: it both applies the chain rule to the sine function and differentiates the inner linear function. Each step must change only one thing, so this step should be split into two separate steps.
Proved: SymPy reduced the difference between the two sides to zero. Checked independently: a separate
method, named above, confirmed it. Checked numerically: the two sides agree at every sampled point, which is evidence,
not proof. Reviewed: a model or a person read it; that is all a sentence can have. Solution by
gemma4:26b, checked 2026-10-09 with SymPy 1.14.0.